Other indication for Dornish law or at least cultural norms come from Gaemon Palehair edict that is said to be influenced by his Dornish stepmother.
Notably, that list is found in World of Ice and Fire not in the history section, but in the Dorne section. So in context, this is very much presented as an example of Dornish laws.
Overall, it seems Dorne ends up with a more drawn out joining process than canon, but ever so slightly less autonomy. In canon it is Dornish law rules in Dorne, period, which presumably the Prince of Dorne, as Prince of Dorne can (in theory, not withstanding the power of his lords) can change as he sees fit. As well, there is no mention here of Dorne collecting its own taxes separate from royal tax collection, though admittedly that is the only time in any source work royal tax collection has been brought up at all, so the omission may also be due to that.
So, yeah, in a way, it's actually better terms for the Realm than canon. It's fine.
It might be a worthwhile project later on to consider what can be 'Kingdom Laws' and what should be reserved as 'Throne Laws'.
There are no separate laws for separate kingdoms. King Jaehaerys saw to that. There is a unified legal code for all of the Realm... except for Dorne, who as a matter of special autonomy stands outside that law. It is
just Dorne. Which in a way is also the problem, as that will raise the ire and envy of others, but well, what can I say. Dorne just is that special
(oh and yeah, de facto the Iron Islands of course also ignore that legal code, but officially they are still covered by it)